A cluster randomized controlled trial (C-RCT) is common in educational intervention studies. Multilevel modelling (MLM) is a dominant analytic method to evaluate treatment effects in a C-RCT. In most MLM applications intended to detect an interaction effect, a single interaction effect (called a conflated effect) is considered instead of level-specific interaction effects in a multilevel design (called unconflated multilevel interaction effects), and the linear interaction effect is modelled. In this paper we present a generalized additive mixed model (GAMM) that allows an unconflated multilevel interaction to be estimated without assuming a prespecified form of the interaction. R code is provided to estimate the model parameters using maximum likelihood estimation and to visualize the nonlinear treatment-by-covariate interaction. The usefulness of the model is illustrated using instructional intervention data from a C-RCT. Results of simulation studies showed that the GAMM outperformed an alternative approach to recover an unconflated logistic multilevel interaction. In addition, the parameter recovery of the GAMM was relatively satisfactory in multilevel designs found in educational intervention studies, except when the number of clusters, cluster sizes, and intraclass correlations were small. When modelling a linear multilevel treatment-by-covariate interaction in the presence of a nonlinear effect, biased estimates (such as overestimated standard errors and overestimated random effect variances) and incorrect predictions of the unconflated multilevel interaction were found.

中文翻译：

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使用广义加性混合模型在集群随机对照试验中对多级非线性协变量治疗进行建模

整群随机对照试验 (C-RCT) 在教育干预研究中很常见。多级建模 (MLM) 是评估 C-RCT 治疗效果的主要分析方法。在大多数旨在检测交互效果的 MLM 应用程序中，考虑单个交互效果（称为混合效果）而不是多级设计中特定于级别的交互效果（称为非混合多级交互）效应），并建模线性交互效应。在本文中，我们提出了一个广义加性混合模型 (GAMM)，它允许在不假设预先指定的交互形式的情况下估计未合并的多级交互。提供 R 代码以使用最大似然估计来估计模型参数，并可视化非线性处理-协变量交互。使用来自 C-RCT 的教学干预数据说明了该模型的有用性。模拟研究的结果表明，GAMM 在恢复未合并的逻辑多级交互方面优于另一种方法。此外，在教育干预研究中发现的多层次设计中，GAMM 的参数恢复相对令人满意，除非当聚类数量、聚类大小、和组内相关性很小。在存在非线性效应的情况下对线性多级协变量交互作用进行建模时，会发现有偏估计（例如高估标准误差和高估随机效应方差）以及对未合并多级交互作用的错误预测。